Theorem nb3grprlem1 27747

Description: Lemma 1 for nb3grpr 27749. (Contributed by Alexander van der Vekens, 15-Oct-2017.) (Revised by AV, 28-Oct-2020.)

Hypotheses

Ref	Expression
nb3grpr.v	⊢ 𝑉 = (Vtx‘𝐺)
nb3grpr.e	⊢ 𝐸 = (Edg‘𝐺)
nb3grpr.g	⊢ (𝜑 → 𝐺 ∈ USGraph)
nb3grpr.t	⊢ (𝜑 → 𝑉 = {𝐴, 𝐵, 𝐶})
nb3grpr.s	⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))

Assertion

Ref	Expression
nb3grprlem1	⊢ (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))

Proof of Theorem nb3grprlem1

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nb3grpr.s	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))
2		prid1g 4696	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑌 → 𝐵 ∈ {𝐵, 𝐶})
3	2	3ad2ant2 1133	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐵 ∈ {𝐵, 𝐶})
4	1, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ {𝐵, 𝐶})
5	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ {𝐵, 𝐶})
6		eleq2 2827	. . . . . . 7 ⊢ ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
7	6	eqcoms 2746	. . . . . 6 ⊢ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
8	7	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
9	5, 8	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ (𝐺 NeighbVtx 𝐴))
10		nb3grpr.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ USGraph)
11		nb3grpr.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
12	11	nbusgreledg 27720	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐵, 𝐴} ∈ 𝐸))
13		prcom 4668	. . . . . . . . 9 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
14	13	a1i 11	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → {𝐵, 𝐴} = {𝐴, 𝐵})
15	14	eleq1d 2823	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → ({𝐵, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
16	12, 15	bitrd 278	. . . . . 6 ⊢ (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
17	10, 16	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
18	17	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
19	9, 18	mpbid 231	. . 3 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐵} ∈ 𝐸)
20		prid2g 4697	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑍 → 𝐶 ∈ {𝐵, 𝐶})
21	20	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 ∈ {𝐵, 𝐶})
22	1, 21	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ {𝐵, 𝐶})
23	22	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ {𝐵, 𝐶})
24		eleq2 2827	. . . . . . 7 ⊢ ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
25	24	eqcoms 2746	. . . . . 6 ⊢ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
26	25	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
27	23, 26	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ (𝐺 NeighbVtx 𝐴))
28	11	nbusgreledg 27720	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐶, 𝐴} ∈ 𝐸))
29		prcom 4668	. . . . . . . . 9 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
30	29	a1i 11	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → {𝐶, 𝐴} = {𝐴, 𝐶})
31	30	eleq1d 2823	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸))
32	28, 31	bitrd 278	. . . . . 6 ⊢ (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
33	10, 32	syl 17	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
34	33	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
35	27, 34	mpbid 231	. . 3 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐶} ∈ 𝐸)
36	19, 35	jca 512	. 2 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))
37		nb3grpr.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
38	37, 11	nbusgr 27716	. . . . 5 ⊢ (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐴) = {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
39	10, 38	syl 17	. . . 4 ⊢ (𝜑 → (𝐺 NeighbVtx 𝐴) = {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
40	39	adantr 481	. . 3 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
41		nb3grpr.t	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 = {𝐴, 𝐵, 𝐶})
42		eleq2 2827	. . . . . . . . . 10 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
43	41, 42	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
44	43	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
45		vex 3436	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
46	45	eltp 4624	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑣 = 𝐴 ∨ 𝑣 = 𝐵 ∨ 𝑣 = 𝐶))
47	11	usgredgne 27573	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → 𝐴 ≠ 𝑣)
48		df-ne 2944	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ≠ 𝑣 ↔ ¬ 𝐴 = 𝑣)
49		pm2.24 124	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 = 𝑣 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
50	49	eqcoms 2746	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
51	50	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝐴 = 𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
52	48, 51	sylbi 216	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ≠ 𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
53	47, 52	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
54	53	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ USGraph → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
55	10, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
56	55	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
57	56	com3r 87	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
58		orc 864	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))
59	58	2a1d 26	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
60		olc 865	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐶 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))
61	60	2a1d 26	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐶 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
62	57, 59, 61	3jaoi 1426	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐴 ∨ 𝑣 = 𝐵 ∨ 𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
63	46, 62	sylbi 216	. . . . . . . . 9 ⊢ (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
64	63	com12 32	. . . . . . . 8 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
65	44, 64	sylbid 239	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
66	65	impd 411	. . . . . 6 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
67		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = 𝐵
68	67	3mix2i 1333	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)
69	1	simp2d 1142	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
70		eltpg 4621	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ 𝑌 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)))
71	69, 70	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)))
72	68, 71	mpbiri 257	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
73	72	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
74		eleq1 2826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
75	74	bicomd 222	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝐵 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
76	75	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
77	73, 76	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
78	42	bicomd 222	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
79	41, 78	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
80	79	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
81	77, 80	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → 𝑣 ∈ 𝑉)
82	81	ex 413	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 = 𝐵 → 𝑣 ∈ 𝑉))
83	82	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐵 → 𝑣 ∈ 𝑉))
84	83	impcom 408	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → 𝑣 ∈ 𝑉)
85		preq2 4670	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = 𝑣 → {𝐴, 𝐵} = {𝐴, 𝑣})
86	85	eleq1d 2823	. . . . . . . . . . . . . 14 ⊢ (𝐵 = 𝑣 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
87	86	eqcoms 2746	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝐵 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
88	87	biimpcd 248	. . . . . . . . . . . 12 ⊢ ({𝐴, 𝐵} ∈ 𝐸 → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
89	88	ad2antrl 725	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
90	89	impcom 408	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → {𝐴, 𝑣} ∈ 𝐸)
91	84, 90	jca 512	. . . . . . . . 9 ⊢ ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸))
92	91	ex 413	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
93		tpid3g 4708	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∈ 𝑍 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
94	93	3ad2ant3 1134	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
95	1, 94	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
96	95	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
97		eleq1 2826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝐶 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
98	97	bicomd 222	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝐶 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
99	98	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
100	96, 99	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
101	79	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
102	100, 101	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → 𝑣 ∈ 𝑉)
103	102	ex 413	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 = 𝐶 → 𝑣 ∈ 𝑉))
104	103	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐶 → 𝑣 ∈ 𝑉))
105	104	impcom 408	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → 𝑣 ∈ 𝑉)
106		preq2 4670	. . . . . . . . . . . . . . 15 ⊢ (𝐶 = 𝑣 → {𝐴, 𝐶} = {𝐴, 𝑣})
107	106	eleq1d 2823	. . . . . . . . . . . . . 14 ⊢ (𝐶 = 𝑣 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
108	107	eqcoms 2746	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝐶 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
109	108	biimpcd 248	. . . . . . . . . . . 12 ⊢ ({𝐴, 𝐶} ∈ 𝐸 → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
110	109	ad2antll 726	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
111	110	impcom 408	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → {𝐴, 𝑣} ∈ 𝐸)
112	105, 111	jca 512	. . . . . . . . 9 ⊢ ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸))
113	112	ex 413	. . . . . . . 8 ⊢ (𝑣 = 𝐶 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
114	92, 113	jaoi 854	. . . . . . 7 ⊢ ((𝑣 = 𝐵 ∨ 𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
115	114	com12 32	. . . . . 6 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣 = 𝐵 ∨ 𝑣 = 𝐶) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
116	66, 115	impbid 211	. . . . 5 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) ↔ (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
117	116	abbidv 2807	. . . 4 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣 ∣ (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)} = {𝑣 ∣ (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)})
118		df-rab 3073	. . . 4 ⊢ {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝑣 ∣ (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)}
119		dfpr2 4580	. . . 4 ⊢ {𝐵, 𝐶} = {𝑣 ∣ (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)}
120	117, 118, 119	3eqtr4g 2803	. . 3 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝐵, 𝐶})
121	40, 120	eqtrd 2778	. 2 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶})
122	36, 121	impbida 798	1 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∨ w3o 1085   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {cab 2715   ≠ wne 2943  {crab 3068  {cpr 4563  {ctp 4565  ‘cfv 6433  (class class class)co 7275  Vtxcvtx 27366  Edgcedg 27417  USGraphcusgr 27519   NeighbVtx cnbgr 27699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-hash 14045  df-edg 27418  df-upgr 27452  df-umgr 27453  df-usgr 27521  df-nbgr 27700

This theorem is referenced by:  nb3grpr  27749  nb3grpr2  27750  nb3gr2nb  27751